equivalent circuit modeling and vibrometry measurements of

Equivalent circuit modeling and vibrometry measurements ofthe Nigerian-origin Udu Utar drum

Brian E. Andersona)

Acoustics Research Group, Department of Physics and Astronomy, Brigham Young University, N283 EyringScience Center, Provo, Utah 84602

C. Beau HiltonDepartment of Humanities, Classics, and Comparative Literature, Brigham Young University,4110 Joseph F. Smith Building, Provo, Utah 84602

Frank GiorginiUdu Incorporated, 4425 County Route 67, Freehold, New York 12431

(Received 10 October 2012; revised 20 December 2012; accepted 14 January 2013)

The Udu drum, sometimes called the water pot drum, is a traditional Nigerian instrument. Musi-

cians who play the Udu exploit its aerophone and idiophone resonances. This paper will discuss an

electrical equivalent circuit model for the Udu Utar, a modern innovation of the traditional Udu, to

predict the low frequency aerophone resonances and will also present scanning laser vibrometer

measurements to determine the mode shapes of the dominant idiophone resonances. These analyses

not only provide an understanding of the unique sound of the Udu instrument but may also be used

by instrument designers to create instruments with resonance frequencies at traditional musical

intervals for the various tones produced and to create musical harmonic ratios. The information,

specifically the laser vibrometry measurements, may also be useful to musicians in knowing the

best places to strike the Udu to excite musical tones. VC 2013 Acoustical Society of America.

[http://dx.doi.org/10.1121/1.4789892]

PACS number(s): 43.75.Hi, 43.20.Ks [TRM] Pages: 1718–1726

I. INTRODUCTION

Originating with the Igbo people of Nigeria, the Udu is

a clay water pot constructed for musical use. It is both an idi-

ophone—an instrument using vibration of the instrument’s

body to produce sound—and an aerophone—an instrument

that uses vibrating columns and cavities of air to produce

sound. The typical Udu retains the traditional Igbo water pot

shape, reminiscent of a gourd, but derives many of its unique

aural qualities from the addition of a hole in the side of the

pot which enables players to modulate the tones produced by

the aerophone. The side-hole water pot drum has uncertain

beginnings, and ascertaining the reasons for punching the

extra hole in the pot in ancient Igboland will be left to

anthropologists. Of its cultural significance, this much is

known: until recent decades, only women made and played

the Udu, relative to their prominent role in Igbo religion;1

the deep bass aerophonic resonances of the Udu were and

are believed to be the “voices of the ancestors,”2 and as such

the Udu plays an important role in religious ceremonies—in

Chinua Achebe’s Things Fall Apart, arguably the most im-

portant and influential Igbo and pan-African piece of litera-

ture, the Udu is mentioned alongside other Igbo percussion

instruments as forming the basis of the rhythmic support for

these ceremonies.3

Frank Giorgini, an artisan working in upstate New York,

is primarily responsible for bringing the Udu into popularity

outside of Nigeria. While studying pottery in Maine, Giorgini

came into contact with Abbas Ahuwan, a Nigerian potter, and

was introduced to the Udu drum. Giorgini began researching

and producing Udus in earnest, eventually traveling to Nigeria

to learn from master Igbo Udu maker Ahuwan.2,4 Artists such

as Sting, Miles Davis, and Alex Acu~na have utilized Giorgi-

ni’s Udus in their music, and, recently, mass production of the

Udu by percussion companies, including Latin Percussion,

Inc., has enabled many musicians outside of Nigeria to learn

and experiment with the drums.

The literature currently extant on the Udu is limited to a

few articles in various music and instrument magazines,

describing the sound, method of playing, and background of

the drum. The most complete of these is the article written

by Giorgini in 1990,2 which includes the history of and his

history with the drums, a brief description of how Udu are

made, Giorgini’s experimentation and innovation, brief qual-

itative description of the link between Helmholtz resonances

and the Udu, and Giorgini’s goals with Udu production. No

in-depth scientific description or modeling of the acoustics

of the Udu has yet been published, and the purpose of this

paper is to expand the current understanding of these unique

instruments by filling this gap.

This paper provides a basis for understanding the sound

radiated from an Udu from both idiophonic and aerophonic

standpoints by making acoustic measurements of an Udu

Utar, one of the mass-produced models from the Giorgini

collaboration with Latin Percussion, Inc. An equivalent

a)Author to whom correspondence should be addressed. Present address:

Los Alamos National Laboratory, Geophysics Group (EES-17), MS D446,

Los Alamos, New Mexico 87545. Electronic mail: [email protected]

1718 J. Acoust. Soc. Am. 133 (3), March 2013 0001-4966/2013/133(3)/1718/9/$30.00 VC 2013 Acoustical Society of America

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.187.97.22 On: Thu, 13 Mar 2014 16:30:38

mailto:[email protected]

circuit model of the aerophone resonances is developed, and,

using scanning laser vibrometry, the modal patterns on the

top surface of the Udu Utar at the idiophone resonance fre-

quencies have been measured. This information is useful

both to musicians and Udu makers as mastery of the instru-

ment and tighter control of production rely on an understand-

ing of what sounds are possible, how to produce them in

playing, and what parameters might be changed when mak-

ing an Udu to meet any particular musical specification.

II. PLAYING THE UDU

The interest of the Udu drum is primarily in the wide

range of sounds that can be coaxed from it. We will qualita-

tively describe aerophonic resonances and how to obtain

them, and then idiophonic resonances and how to obtain

them. By combining these techniques, a vast sound palette is

achievable. Though the Udu Utar is the specific subject of

our research (shown in Fig. 1), the sounds obtained are typi-

cal of all Udu drums and similar techniques can be applied.

Note that in Sec. I, we refer to the side hole as being a hole

cut into the large cavity of the pot. However, from this point

on we will refer to the hole in the cavity of the pot as the

“top hole” and the hole out of the neck of the pot as the “side

hole” with reference to their respective positions in the photo

in Fig. 1.

The signature sounds of the Udu are the deep bass aero-

phonic resonances and terminal octave slide. By using the

palms of the hands to strike and cover either of the openings,

low bass tones are achieved. The side hole is typically more

responsive to this action with a much stronger response

obtained. When the hand strikes a hole and immediately

releases, the tone quickly slides up nearly an octave before

dissipating.

All of the tone gradations between the lowest note and

the highest note can be achieved as well. Closing the top

hole and striking the side hole achieves the low tone, and as

the player raises one side of the palm on the top hole and

continues to strike the side hole the tone raises in frequency.

As the angle between the palm and the top hole increases to

perpendicular, the pitch increases to the tone nearly an

octave higher. Acoustically the player is dynamically chang-

ing the end correction for the top hole, and therefore the

amount of acoustic mass loading at that end of the Udu.

When the palm covers the top hole the acoustic mass goes to

infinity, but as the palm opens up the top hole, from closed

to all the way open, the acoustic mass decreases to the finite

value determined by the end correction for the acoustical re-

actance presented to that hole. This will be further analyzed

in Sec. III C .

Many other sounds can also be made. The player can

choose to only cover part of either hole when striking and

thus achieve many different tones; covering the side hole

and striking the top hole without completely closing it off

can produce very low tones; and a “swoop” sound, similar to

a dripping faucet, can be made by covering the side hole and

striking the top hole with the palm, completely covering it

and immediately releasing.

The various idiophonic resonances of the Udu are

excited by tapping different parts of the body of the Udu

yielding different dominant tones. These tones are higher in

frequency than the aerophonic tones and often ring for some

time. The duration of these tones depends on the quality of

the Udu (the Utar is a fired clay instrument giving its walls a

high degree of stiffness and thus a high quality factor) and

the method of holding the Udu. An Udu typically rests in the

player’s lap, but this tends to dampen the vibration of the

clay. To raise the Udu off of the player’s lap, straw rings that

the Udu rests on are traditionally used, and mounts of foam-

covered tubular steel are also in use. Both of these methods

allow the Udu to vibrate with less damping and longer dura-

tion for the idiophonic resonances.

Any type of tap, slap, or slide that the player can think

of is considered appropriate technique, but typical playing

method consists of tapping with the pads and inner joints of

one’s fingers. Using knuckles, nails, and full hand slaps is

also common, and persons who have played conga or tabla

generally have success transferring the techniques they al-

ready know. Non-traditional tools such as mallets are also

used on occasion, depending on the musical situation.

Finally, it should be mentioned that Udu drums are

sometimes filled partially with water for playing, which

opens many more possibilities for aural effects and seems to

improve the sound projection in some Udu. These effects

will partly be considered in Sec. III E .

III. AEROPHONE RESONANCES

A. Acoustic measurements

There are three main aerophone, or acoustical, resonan-

ces that can be easily made on the Udu Utar drum, though

these three resonances may be excited in five principal ways.

Three of the excitations can be made by striking the side

hole of the drum and two more can be made by striking the

top hole. The first acoustical resonance can be excited with

the side closed and the top open (SCTO) by striking the side

hole and leaving the hand there to cover the hole, all the

while with the top hole remaining open. The second acousti-

cal resonance can be excited with the side open and the top

closed (SOTC) by striking the side hole and releasing the

hand quickly to leave the side hole open, all the while with

the top hole being closed by the player’s other hand. The

third acoustical resonance can be excited with the side open

and the top open (SOTO) by striking the side hole and

releasing the hand quickly to leave the side hole open, all theFIG. 1. (Color online) Photograph of the Udu Utar drum used in this study.

J. Acoust. Soc. Am., Vol. 133, No. 3, March 2013 Anderson et al.: Analysis of the Udu drum 1719


while with the top hole remaining open. The second acousti-

cal resonance can also be excited with the top closed and the

side open (TCSO) by striking the top hole and leaving the

hand there to cover the hole, all the while with the side hole

remaining open. The third acoustical resonance can also be

excited with the top open and the side open (TOSO) by strik-

ing the top hole and releasing the hand quickly to leave the

top hole open, all the while with the side hole remaining

open. A significantly audible response at the first acoustical

resonance frequency cannot be made by striking the top hole

and releasing the hand quickly to leave the top hole open, all

the while with the side hole being closed by the player’s

other hand.

The five ways to excite the resonances of the Udu Utar

drum were individually recorded in an anechoic chamber. A

type-1, calibrated, free-field microphone was placed 1 m

from the Udu (PCB 377A02 microphone and Larson Davis

PRM426 preamplifier) to record the spectra in conjunction

with a Hewlett Packard 35670 dynamic signal analyzer. The

resonances were produced 3 or 4 times for each recording

and the analyzer averaged the results. The resulting spectra

are displayed in Fig. 2. Note that the resonances for TOSO

and SOTO are at the same frequency as should be expected

since the boundary conditions are the same once the hand

has excited the instrument and subsequently been removed

from the instrument. Note also that the resonances for SOTC

and TCSO are also at the same frequency since the boundary

conditions are again the same after the initial excitation. The

three aerophone resonance frequencies are 73, 113, and

137 Hz, and essentially correspond to the musical notes D2,

A2, and C#3 .

As observed from Fig. 2, the Udu Utar essentially has

three distinct aerophone resonances. The frequencies of

these resonances are nearly spaced according the musical

intervals of a fifth and a seventh relative to the lowest reso-

nance frequency. Perhaps one of the more interesting acous-

tic features of the sounds produced by the Udu Utar is the

varying tone produced by adjusting the opening of one hole

while striking the other. This feature is produced by dynami-

cally modifying the degree to which one hand covers the top

hole as the side hole is struck and released (ranging from

SOTC 73 Hz to SOTO 137 Hz). The musician is thus dynam-

ically changing the end correction (the inertia and resistance

at the opening of the top hole) of the top hole. Figure 3 dis-

plays a spectrogram from a recording of a player dynami-

cally varying the top hole opening while striking and

releasing the side hole opening. This figure will be discussed

further later on.

B. Advanced equivalent circuit modeling

The aerophone resonances of the Udu Utar drum are

low enough in frequency such that their wavelengths are

large compared to the internal dimensions of the drum. For

example, the highest aerophone resonance (at present con-

sideration) of 137 Hz corresponds to a wavelength of 2.50 m,

whereas the largest internal dimension is approximately

30 cm. For reference, the length of the neck is 14.6 cm and

the cavity volume is 9.25 L (9.25� 10�3 m3). Large wave-

lengths compared to the size of the resonator allow the

aerophone resonances of the instrument to be modeled with

one-dimensional, electrical equivalent circuit modeling

techniques. The advantage of these techniques is that the

complicated curvature of the instrument need not be known

FIG. 2. (Color online) Frequency spectra of the sound produced by acoustic

resonances of the Udu Utar drum. (a) Side hole excitations. (b) Top hole

excitations.

FIG. 3. (Color online) Spectrogram of the sound produced by the Udu Utar

drum as the instrument is struck and released at the side hole multiple times

while the alternate hand is used to dynamically change the end condition on

the top hole from a closed condition to an open condition, back to a closed

condition (this cycle is repeated twice).

1720 J. Acoust. Soc. Am., Vol. 133, No. 3, March 2013 Anderson et al.: Analysis of the Udu drum


exactly. Using the rules of constructing equivalent circuits,

one can then create an electrical circuit diagram that may be

solved for impedance quantities, which yield the resonance

frequencies of the Udu.

Figure 4(a) displays the simplified-geometry drawing of

the Udu Utar used in the equivalent circuit modeling (with the

side hole on the left). The various parts of the instrument are

then represented as electrical components in the so-called

acoustic impedance domain equivalent circuit. The large cav-

ity, labeled “C” in the drawing, of the Utar is modeled using a

two-port T-network (the expressions used in T-networks are

given by Mason in Ref. 5). The length for the network model-

ing the large cavity was determined from a measurement of

the distance between the center of the top hole and the center

of the junction between the neck and the cavity. The surface

area of the large cavity was determined by dividing the vol-

ume of the cavity (measured by filling the Udu cavity up with

water and then measuring the volume of the water) by this

measured cavity length. The neck of the instrument, labeled

“N1� N4,” whose geometry flares out towards the side hole

and then abruptly constricts to the side hole opening, is mod-

eled with four T-networks in series, using four segments of

the neck length and using the average surface area of those

segments. The junction between the neck and the cavity is

modeled as a mass, MAJ ¼ 8q0=½3p2rN4Hðb=aÞ� (see Table I

for definitions of the symbols used here and throughout the

paper), according to Karal’s Eq. (52), with Hðb=aÞ ¼ 0:7.6

The interface between the cavity and the top hole is mo-

deled as an orifice with a resistance, RAOT ¼ lnð4rT=tÞffiffiffiffiffiffiffiffiffiffiffiffiffiffi2q0xlp

=ð4STÞ, and a mass, MAOT ¼ q0rT=4, according to

the expressions given by Morse and Ingard7 [Eq. (9.1.23)] and

Pierce8 [Eq. (7-5.10)], respectively. The interface between the

side hole and the neck is also modeled as an orifice with a re-

sistance, RAOS ¼ lnð4rS=tÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2q0xlp

=ð4SSÞ, and a mass,

MAOS ¼ q0rS=4, using those same equations. The radiation

impedance loading at the outlet of the top hole is modeled

with resistance, RART ¼ q0cðkrTÞ2=ð2STÞ, and mass loading

(related to the reactance), MART ¼ 8q0rT=ð3pSTÞ, values by

approximating this opening as baffled (or flanged) using Eq.

(10.2.10) of Kinsler et al.9 The radiation impedance loading

at the outlet of the side hole is also modeled with resistance,

RARS ¼ q0cðkrSÞ2=ð2SSÞ, and mass loading, MARS ¼ 8q0rS=ð3pSSÞ, values by also approximating this opening as being

baffled. Figure 4(b) shows the equivalent circuit for the Udu

Utar. The impedances represented in Fig. 4(b) are

Z1 ¼ RARS þ RAOS þ jxðMARS þMAOSÞ; (1)

Z2 ¼ jq0c

SN1

tankl1

2

� �; (2)

Z3 ¼ �jq0c

SN1

cscðkl1Þ; (3)

Z4 ¼ jq0c

SN1

tankl1

2

� �þ j

q0c

SN2

tankl2

2

� �; (4)

Z5 ¼ �jq0c

SN2

cscðkl2Þ; (5)

Z6 ¼ jq0c

SN2

tankl2

2

� �þ j

q0c

SN3

tankl3

2

� �; (6)

Z7 ¼ �jq0c

SN3

cscðkl3Þ; (7)

Z8 ¼ jq0c

SN3

tankl3

2

� �þ j

q0c

SN4

tankl4

2

� �; (8)

Z9 ¼ �jq0c

SN4

cscðkl4Þ; (9)

Z10 ¼ jq0c

SN4

tankl4

2

� �þ jxMAJ þ j

q0c

SCtan

klC

2

� �; (10)

Z11 ¼ �jq0c

SCcscðklCÞ; (11)

FIG. 4. Acoustical domain equivalent circuit model for the acoustic

resonances of the Udu Utar drum. (a) Drawing of acoustical model. (b) The

equivalent electrical circuit.

TABLE I. Symbol explanation. The angular frequency and acoustic wave-

number vary according to the frequency range desired. The other quantities

listed are measured (from length measurements of the Udu Utar) or esti-

mated from location specific information such as altitude and temperature

(as in the case of the speed of sound, atmospheric density, and coefficient of

viscosity).

Symbol Unit Description

j — imaginary number,ffiffiffiffiffiffiffi�1p

x rad/s angular frequency

c m/s speed of sound in air¼ 340 m/s in Provo, UT

k 1/m acoustic wavenumber

q0 kg/m3 atmospheric density¼ 1.01 kg/m3 in Provo, UT

l Pa�s coefficient of viscosity¼ 1.85� 10�5 P as

t m thickness of Udu Utar wall at hole exits¼ 0.006 m

lNX m Nth length of neck segment¼ [4.1, 3.1, 3.6, 2.0]� 10�2 m

rNX m Nth radius of neck segment¼ [4.7, 4.0, 3.3, 2.9]� 10�2 m

SNX m2 Nth surface area of neck

segment¼ [7.1, 4.9, 3.3, 2.5]� 10�2 m2

rS m radius of side hole opening¼ 0.035 m

SS m2 surface area of side hole opening¼ 0.0038 m2

rT m radius of top hole opening¼ 0.025 m

ST m2 surface area of top hole opening¼ 0.0020 m2

lC m measured length of large cavity, from neck to top

hole¼ 0.229 m

SC m2 effective surface area of large cavity¼ 0.041 m2



Z12 ¼ jq0c

SCtan

klC

2

� �; (12)

Z13 ¼ RART þ RAOT þ jxðMART þMAOTÞ: (13)

The SOTO resonance, caused when the hand strikes the side

hole and releases quickly, is found by determining the im-

pedance at the terminals labeled “a” in Fig. 4(b), then finding

the point at which the imaginary part of this impedance (the

reactance) goes to zero. The striking excitation creates a

pulse of energy that travels down the neck (represented by

impedances Z2 through the first term of Z10) and reaches the

junction (represented by the second term in Z10) at the acous-

tic cavity. Some of the energy resonates within the cavity

(represented by the third term in Z10 through Z12) and some

escapes through the top hole (represented by Z13). Some of

the initial energy also escapes out the side hole (represented

by Z1) without traveling down the neck. The SOTO react-

ance goes to zero at fSOTO ¼ 141:3 Hz and represents a 3.1%

error from the measured value of 137 Hz [see Fig. 2(a)].

The SOTC resonance is found by setting Z12 ¼ Z13 ¼ 0

and determining the reactance presented to the terminals

labeled “a” in Fig. 4(b). This resonance is similar to that of

SOTO except that energy is prevented from escaping from

the top hole. The SOTC reactance goes to zero at fSOTC

¼ 75:8 Hz and represents a 3.8% error from the measured

value of 73 Hz [see Fig. 2(a)].

The SCTO resonance is found by setting Z1 ¼ Z2 ¼ 0

and determining the reactance presented to the terminals

labeled “b” in Fig. 4(b). The SCTO resonance prevents energy

from radiating out of the side hole. The SCTO reactance goes

to zero at fSCTO ¼ 113:8 Hz and represents a 0.7% error from

the measured value of 113 Hz [see Fig. 2(a)].

The TOSO resonance is found by determining the react-

ance presented to the terminals labeled “d” in Fig. 4(b). The

TOSO excitation should present the same resonator as the

SOTO excitation. Indeed the TOSO reactance goes to zero at

fTOSO ¼ 141:2 Hz (nearly identical to that found for fSOTO)

and represents a 3.0% error from the measured value of

137 Hz [see Fig. 2(b)].

The TCSO resonance is found by setting Z12 ¼ Z13 ¼ 0

and determining the reactance presented to the terminals la-

beled “c” in Fig. 4(b). The TCSO excitation should present

the same resonator as the SOTC excitation. Indeed the

TCSO reactance goes to zero at fTCSO ¼ 75:8 Hz (identical

to that found for fSOTC) and represents a 3.8% error from the

measured value of 74 Hz [see Fig. 2(b)].

C. Discussion

A preliminary simplified circuit model, which did not

divide up the neck into segments or use T-network circuits

anywhere in the model, constructed by the authors, yielded

resonance frequencies that were within 9.3% error.10 With the

equivalent circuit model considered here the resonance fre-

quencies are predicted to within 3.8% error (average of 2.9%

error). A circuit model with further complexities, such as

dividing up the neck into smaller sections and dividing the

large cavity up into sections could result in even higher accu-

racy. Perhaps the principal advantage of using the equivalent

circuit models is that one may adjust an existing geometrical

design of an instrument to optimally produce desired musical

intervals, or one may design an entirely new type of Udu

drum with desired musical intervals. For example, one could

adjust the surface areas of the holes, length of the neck, and/or

cavity volume of the Udu Utar in an attempt to adjust the mu-

sical intervals to the desired ones. With an accuracy of 3.8%,

one may not be able to construct a new design with confi-

dence that it will have the exact musical ratios desired. How-

ever, if one models an existing design and wishes to tweak the

musical ratios, then the model may be used to determine rela-

tive changes in lengths, areas, or volumes required to change

the musical ratios by the desired amounts.

As discussed in Sec. II, when one seals off the top hole

with one hand and strikes (and releases) the side hole, the

deep fSOTC ¼ 73 Hz resonance is excited. However, as one

slowly uncovers the top hole while simultaneously continu-

ing to strike the side hole, the pitch increases. When the

hand on the top hole is completely removed from covering

the hole, a strike at the side hole produces a pitch of

fSOTO ¼ 137 Hz. A spectrogram of a recorded demonstration

of a couple full cycles of this dynamic pitch variation may

be found in Fig. 3. A similar type of dynamic pitch variation

may also be demonstrated using an pipe where one repeat-

edly strikes one end of the pipe and varies the degree to

which the other end is sealed off, which is akin to exciting

the open-closed fundamental resonance of the pipe up to the

open-open resonance of the pipe. In the case of the Udu, the

MART and MAOT values increase from an open top hole condi-

tion to infinity when the top hole is closed, which results in

Z13 !1 and hence prevents current from flowing into the

Z12; Z13 branch of the circuit.

D. Simplified dual opening Helmholtz resonator

A simplified model of the Udu is dual opening Helm-

holtz resonator represented by an acoustic compliance,

C ¼ V=½q0c2�, where C is the volume of the cavity, repre-

senting the cavity with two acoustic masses, M1 ¼ q0l1=S1

and M2 ¼ q0l2=S2, where l and S represent the length and

surface area of the “pipe” opening, respectively, representing

each of the two openings as shown in Fig. 5(a). For a basic

two-opening Helmholtz resonator one would expect the Py-

thagorean relationship for the resonance with both openings

exposed, f12 ¼ ð1=2pÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðM1 þM2Þ=ðM1M2CÞ

p, to the two

resonance frequencies produced when each of the two open-

ings are individually sealed off, f1 ¼ ð1=2pÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=ðM1CÞ

pand

f2 ¼ ð1=2pÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=ðM2CÞ

p, such that f12 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffif 21 þ f 2

2

p. The f12

resonance is found by determining the impedance at the ter-

minals labeled “a” in Fig. 5(b) with the terminals labeled “b”

connected (short circuited) and by determining the frequency

at which the reactance of this impedance equals zero. The f1

resonance is found by determining the impedance at the ter-

minals labeled “a” in Fig. 5(b) with the terminals labeled “b”

left open and by determining the frequency at which the re-

actance of this impedance equals zero. The f2 resonance is

found by determining the impedance at the terminals labeled

“b” in Fig. 5(b) with the terminals labeled “a” left open and



by determining the frequency at which the reactance of this

impedance equals zero.

If we define a quantity a ¼ M1=M2 ¼ ðl1S2Þ=ðl2S1Þ we

can plot the frequency ratios f2=f1 ¼ffiffiffiap

and f12=f1 ¼ffiffiffiffiffiffiffiffiffiffiffiaþ 1p

so that we can design a simplified Udu to have musical inter-

vals for these frequency ratios. Note that if a ¼ 0 then

f12 ¼ f1, if a ¼ 1 then f2 ¼ f1, and if a ¼ 1 then f12 ¼ f2,

none of which are musically useful utilizations of both open-

ings. Here we assume that M1 > M2 and that one would start

by choosing a fundamental frequency f1 from which to design

an Udu cavity and one hole opening. One then might wish to

view the relationships of the frequency ratios defined above in

order to design the second hole opening as shown in Fig. 6.

Note that if one selects a ¼ 1:25 the frequency ratios result in

musical intervals of a second and a fifth relative to f1, or if

one selects a ¼ 1:78 the frequency ratios result in musical

intervals of a fourth and a sixth. Further note that these fre-

quency ratios are independent of the cavity volume, though

the values for the frequencies themselves depend on the cavity

volume. Figure 6 may be used to obtain a quick, rough esti-

mate of the resonance frequencies that will result from a new

type of Udu design based on the choice of lengths and surface

areas of the ports and the volume of the cavity.

The Udu Utar resonance frequencies do not exactly

agree with the Pythagorean relationship of the simplified

model with, fSOTO ¼ 137 Hz whereasffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2SOTC þ f 2

SCTO

q

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi732 þ 1132

pHz ¼ 134:5 Hz. This �1.8% error departure

of this relationship for the Udu Utar is not large but the

authors feel that this small departure is perhaps by chance.

There are competing complexities in the differences between

an idealized dual port Helmholtz resonator and the Utar that

one encounters as one improves the equivalent circuit model

shown in Fig. 6(b). Some of these complexities include the

acoustic resistances in the advanced equivalent circuit model

(which do in fact alter the resonance frequency values), the

radiation mass loadings, and also the effects of orifice type

impedance loadings presented by the relatively small hole

openings on the Utar. The reader should note that despite the

decent agreement in the relationship of the resonance fre-

quencies of the Utar with the simplified model, the dual port

Helmholtz resonator does not predict the actual resonance

frequency values of the Utar to even within 100% error accu-

racy, with predicted values of f1 ¼ 93 Hz, f2 ¼ 243 Hz, and

f12 ¼ 268 Hz (compared to values of 73, 113, and 137 Hz),

underscoring the need for the advanced circuit model.

E. Adjusting the model

In order to test the ability of the equivalent circuit model

to predict changes to an Udu design we add water to the

large cavity of the Utar as is sometimes done by musicians.

When 1.00 L of water is added, we measure two of the reso-

nance frequencies to shift up to values of fSCTO ¼ 116:5 Hz

and fSOTO ¼ 143:5 Hz. The effective length between the

neck junction and the top hole, lC, remains the same when

water is added but we must adjust the effective surface area

of the cavity. Since the volume of the cavity, VC ¼ SClC, we

change SC from a value of 0.041 m2 to a value of 0.036 m2 to

account for the decrease in volume of the cavity by 1.00 L.

The equivalent circuit model then predicts that fSCTO

¼ 119:3 Hz and fSOTO ¼ 149:1 Hz resulting in percent errors

of 2.4% and 3.9%, respectively.

When 2.00 L of water are added, we shift SC to a value

of 0.032 m2. The measured resonance frequencies with 2.00 L

of water added shift upward further to values of fSCTO

¼ 123:5 Hz and fSOTO ¼ 151:5 Hz. We then predict resonance

frequencies of fSCTO ¼ 125:6 Hz and fSOTO ¼ 158:3 Hz with

the equivalent circuit model. This model is off by 1.7 and

4.5 %, respectively.

Thus the model’s predicted resonance frequencies with

added water in the cavity are not much larger than the model

without the water added. While we may not be able to pre-

dict the actual frequency values with accuracy better than

4%, we can use the model to predict the relative resulting

change. One way to analyze the model’s ability to make rela-

tive predictions is to determine the measured shifts in reso-

nance frequencies as a ratio when water is added and

FIG. 5. Simplified Utar acoustical domain equivalent circuit model for the

acoustic resonances of a dual port Helmholtz resonator. (a) Drawing of the

simplified Utar. (b) The equivalent electrical circuit.

FIG. 6. (Color online) Plot of the resonance frequency ratio f=f1 versus the

ratio of the acoustical masses of the two ports, a ¼ M1=M2, for a simplified

model of an Utar (a dual Helmholtz resonator). This plot may be used to cre-

ate an initial design for an Udu like instrument.



compare the measured shifts to the predicted shifts we calcu-

late using the model when water is added. The shifts in the

measured resonance frequencies when the 1.00 L of water is

added are ratios of f 1 LSCTO=fSCTO ¼ 1:03 and f 1 L

SOTO=fSOTO

¼ 1:04, and when 2.00 L of water is added the measured

ratios are f 2 LSCTO=fSCTO ¼ 1:09 and f 2 L

SCTO=fSCTO ¼ 1:11. The

corresponding shifts predicted by the model for 1.00 L of

added water are f 1 LSCTO=fSCTO ¼ 1:05 and f 1 L

SCTO=fSCTO ¼ 1:06,

and are f 1 LSCTO=fSCTO ¼ 1:10 and f 1 L

SCTO=fSCTO ¼ 1:12 when

2.00 L of water is added. Thus the average measured shifts are

1.04 for 1 L and 1.10 for 2 L while the average calculated shifts

are 1.05 for 1 L and 1.11 for 2 L, which is good agreement.

IV. IDIOPHONE RESONANCES

A. Acoustic measurements

As discussed in Sec. II, the Udu is not just an aerophone

instrument but is also an idiophone instrument. A musician

may take advantage of the rich amount of idiophone sounds

by striking the Udu’s surface in various places to excite dif-

ferent relative amounts of various structural resonances of

the instrument’s body. One of the more pronounced and mu-

sical idiophone sounds of the Udu Utar is made by striking

the lip of the end of the instrument where the side hole is

located. The resulting tone is similar to that of a cowbell

tone. Acoustic recordings of the idiophone resonances pro-

duced by the Udu Utar were made in the anechoic chamber

using the same equipment described in Sec. III.

Figure 7 displays measured spectra produced when the

Utar is struck in three different locations: at the cowbell

location described above, a couple centimeters west (left) of

the top hole, and a couple centimeters north (above) of the

top hole. Note that in the spectrum for the cowbell tone that

the lowest dominant partial is at 556 Hz (approximately C#5 )

and there exists nearly harmonic partials at 1108 Hz (approx-

imately C#6 ) and at 1644 Hz (approximately G#

6 ). These

nearly harmonic partials represent ratios of 1.99 and 2.96,

respectively, and thus are slightly flat relative to perfect har-

monic ratios. Note also the partial at 970 Hz (closet to B5),

which is excited strongly by striking near the top hole (with

a stronger response when striking to the west of the top

hole). This partial has a near harmonic located at 1898 Hz (a

ratio of 1.96 relative to the fundamental, nearly an octave

above 970 Hz). Thus when one needs more cowbell and

strikes the lip of the side hole, a nearly harmonic tone is pro-

duced with a fundamental frequency of 556 Hz, while the

response at 970 Hz is greatly reduced. When one strikes near

the top hole a higher pitched tone results with a fundamental

frequency of 970 Hz, while the response at 556 Hz is greatly

reduced. Hence the striking location greatly influences the

pitch and tonal quality of the idiophone resonances.

B. Scanning laser vibrometer measurements

In order to better understand the structural modes of the

Udu Utar that contribute to the tones described in Sec. IV A ,

a Polytec Scanning Laser Doppler Vibrometer (SLDV) PSV-

400 was employed. A LDS Shaker V203, with the shaker’s

stinger affixed with bees wax to the underneath side of the

side hole lip, was used to drive the Udu at selected resonance

frequencies determined from Fig. 7 with a peak to peak volt-

age amplitude of 500 mV. The sensitivity of the SLDV was

adjusted as appropriate to optimize the digitization of the

acquired signals. A 3200 Hz bandwidth with 1600 lines of re-

solution was used along with five complex FFT averages, and

a 20 Hz high pass filter. A rectangular grid of scan points was

made up of 391 total points (23 by 17) with 218 of those

points on the surface of the Utar. Small pieces of reflective

tape were placed at each of the scan point locations to opti-

mize the reflectivity of the incident laser signal, and to

account for the surface contour variations away from normal

incidence. The Udu was driven at frequencies of 556 (C#5 ),

970 (B5), 1108 (C#6 ), 1378 (F6), 1470 (F#

6 ), and 1644 (G#6 ) Hz

corresponding to most of the dominant resonance frequency

peaks in Fig. 7 (the peaks denoted with arrows in Fig. 7).

Figure 8 displays images obtained from the six SLDV

measurements. These images represent snapshots in time of

the spatial distribution of the surface velocity when the

mode reached a maximum in amplitude. Note that while

considered not important here (we are interested in the

modal shapes only), the velocities displayed in Fig. 8 are

only a component of the total out of plane surface velocity at

each scan point (a component in the direction of the laser),

since not all scan points were on surfaces normal to the

laser’s incidence angle. Lines have been drawn onto these

modal images to aid visualization of the nodal lines, along

with þ and � symbols to denote antinodes of relative phases

of positive and negative, respectively. The instantaneous

velocities are given in units of mm/s. Note that the mode

shapes indicate why different modes are more or less

strongly excited when the Udu is struck at different loca-

tions. For example, the 566 Hz mode [Fig. 8(a)], is strongly

excited when struck at the side hole lip due to the presence

of the antinode, while it is minimally excited when struck

just west of the top hole and hardly excited when struck just

north of the top hole, due to the proximity of these striking

locations to a node (refer to Fig. 7 to verify the relative

strengths of exciting the 566 Hz mode at these striking loca-

tions). A similar visual analysis can be made for each of the

modes displayed in Fig. 8, and their corresponding levels in

the sound spectra displayed in Fig. 7, to note their dependen-

cies on striking location and the proximity of the striking

location to the nodes.

FIG. 7. (Color online) Frequency spectra of the sound produced by idio-

phone resonances of the Udu Utar drum: (a) when struck at the cowbell, (b)

when struck north of the top hole, and (c) when struck west of the top hole

(north and west striking locations are identified in Fig. 1).



C. Discussion

Knowledge of the modal patterns of the Udu Utar with

their corresponding resonance frequencies can assist the

musician in learning where to strike the instrument to

achieve tones with harmonic overtones. A player typically

must find sweet spots, or striking locations that produce mu-

sical tones, on the instrument through practice.

The designer of this particular Udu Utar drum spent

2–4 years designing and developing new Udu designs, one

of which was the Utar, that “sounded right” to him based on

years of experience in both playing and designing Udu

drums. Designers do not currently utilize models or measure-

ments to shape the drum and instead rely on their experience

with various designs. Designers of the instrument can also

benefit from this modal analysis information as they can

potentially modify future designs by adding or removing

some of the wall material in certain locations in order to tune

the instrument as is commonly done for xylophones and

marimbas when their bars are undercut to achieve nearly har-

monic tuning of upper partial frequencies. It is interesting to

note that the designer of the Udu Utar was able to produce a

handmade instrument that possesses musically spaced reso-

nance frequencies without the use of a model.

V. CONCLUSIONS

It has been shown that the deep aerophone resonances

of the Udu Utar drum may be predicted using electrical

equivalent circuit modeling techniques. This may aid an

instrument designer in adjusting the design of an Udu drum

to result in desired musical intervals for its principal aero-

phone resonances. The unique ability to produce the full

spectrum of tones between the lowest and highest principal

resonances by varying the end condition of one of the holes

has also been explained. A simplified model of the Utar has

been shown, which may be used as a rough starting point for

designing a newly shaped Udu. Finally, an example has been

given when the model may be adjusted to predict relative

changes in the Udu’s tones when the instrument’s geometry

is changed.

The idiophone resonance frequencies of the Udu Utar

have been found to occur nearly at musical intervals. The

modes that correspond to these resonance frequencies have

been measured with a scanning laser vibrometer. Visual

analysis of the modal patterns allows one to determine the

strength of these resonance frequencies in the radiation spec-

tra. Knowledge of these modal patterns may aid a musician

in learning how to produce musical tones with harmonic

upper partials, and they may aid an instrument designer in

determining how the thickness of the Udu walls may be

slightly modified to fine tune these resonance frequencies.

ACKNOWLEDGMENTS

We note that the word “Udu” is a registered trademark

in the United States owned by Frank Giorgini. The authors

FIG. 8. (Color online) Modal distri-

butions of six selected idiophone

modes at (a) 556, (b) 970, (c) 1108,

(d) 1378, (e) 1470, and (f) 1644 Hz

of the Udu Utar drum. Amplitudes

are displayed on a gray scale (or

color online) and are expressed in

units of mm/s.



also wish to thank Timothy Leishman for circuit modeling

suggestions. They also wish to thank Hillary Jones, Benja-

min Christensen, and Blaine Harker for their assistance in

making some of the measurements used in this paper.

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Freehold, NY, 2009), Vol. 1.

5W. P. Mason, Electromechanical Transducers and Wave Filters (Van

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constrictions of circular cross section,” J. Acoust. Soc. Am. 25(2), 327–

334 (1953).7P. M. Morse and K. U. Ingard, Theoretical Acoustics (Princeton University

Press, Princeton, 1986), pp. 480–482.8A. D. Pierce, Acoustics: An Introduction to Its Physical Principles and Appli-cations (Acoustical Society of America, Woodbury, NY, 1989), pp. 336–341.

9L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamentalsof Acoustics, 4th ed. (Wiley, New York, 2000), pp. 272–284.

10B. Hilton, B. E. Anderson, and H. Jones, “Equivalent circuit modeling and

vibrometry analysis of the Udu Utar Nigerian drum,” J. Acoust. Soc. Am.

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